I've seen this question asked a few times, and I'm definitely in the minority on this, which is fine. Everyone has their own opinion.
From personal experience, My brother and I were 17/15 and 18/16, respectively, for two of our trips. It never even occured to us to ask to go off on our own, unless you count when we went through the line at Space Mountain by ourselves (My parents hung around in a shop in Tomorrowland or did something else nearby. They were there when we came out.). We were there as a family, so we did things together as a family.
I don't even remember there being an instance where we were upset because we had to wait for someone. We all liked (and disliked) the same rides (for the most part). At 15, ToT had recently opened. I wanted to ride, but no one else did. We went through the que together, and everyone else took the chicken elevator back down.
What will I do? My kids will be 4 1/2 years apart. I definitely can't see sending someone younger than 16 off by themselves, so unless we go when the kids are 16 & 20, I doubt that they'd be going off by themselves together. Of course, with DD being so much older, I might do 15 & 19, but definitely not younger than 15. There are too many issues that can come from that, and I'm not just talking about the person hanging around with bad intentions. Even the best behaved kids can get into trouble without parents present (okay, sometimes even with parents present, but the chances are lower).
As a whole, I'm just not comfortable about it. Many people are, and that's fine, too. I remember reading a comment from someone who had let their 12 yr old park hop by themselves. It's not something that I would do, but ultimately it comes down to the parent. If you honestly feel that it would be okay, and you don't have any hesitation about it, I'd say go for it. Everyone knows the stuff that CAN go wrong, and we all know that, on average, none of that stuff does.
It sounds like you're pretty leary about it. If that's the case, I would say don't do it. Would you be able to relax and have a good time, or would you be too concerned with how they are doing? If they don't go off by themselves, it won't be the end of the world. This is one of those instances where just because it's right for other families, doesn't mean that it's right for your family.